WEBVTT
00:00:07.930 --> 00:00:13.530
OK, here we go with, quiz
review for the third quiz that's
00:00:13.530 --> 00:00:15.910
coming on Friday.
00:00:15.910 --> 00:00:19.040
So, one key point
is that the quiz
00:00:19.040 --> 00:00:25.120
covers through chapter six.
00:00:25.120 --> 00:00:27.580
Chapter seven on
linear transformations
00:00:27.580 --> 00:00:31.930
will appear on the final
exam, but not on the quiz.
00:00:31.930 --> 00:00:35.540
So I won't review linear
transformations today,
00:00:35.540 --> 00:00:39.720
but they'll come into the
full course review on the very
00:00:39.720 --> 00:00:41.350
last lecture.
00:00:41.350 --> 00:00:44.120
So today, I'm
reviewing chapter six,
00:00:44.120 --> 00:00:46.330
and I'm going to
take some old exams,
00:00:46.330 --> 00:00:48.990
and I'm always ready
to answer questions.
00:00:48.990 --> 00:00:54.360
And I thought, kind of help
our memories if I write down
00:00:54.360 --> 00:01:00.350
the main topics in chapter six.
00:01:00.350 --> 00:01:04.019
So, already, on
the previous quiz,
00:01:04.019 --> 00:01:07.520
we knew how to find
eigenvalues and eigenvectors.
00:01:07.520 --> 00:01:13.480
Well, we knew how to find them
by that determinant of A minus
00:01:13.480 --> 00:01:15.390
lambda I equals zero.
00:01:15.390 --> 00:01:17.280
But, of course, there
could be shortcuts.
00:01:17.280 --> 00:01:20.030
There could be, like,
useful information
00:01:20.030 --> 00:01:27.190
about the eigenvalues that
we can speed things up with.
00:01:27.190 --> 00:01:27.830
OK.
00:01:27.830 --> 00:01:32.300
Then, the new stuff starts out
with a differential equation,
00:01:32.300 --> 00:01:34.680
so I'll do a problem.
00:01:34.680 --> 00:01:36.810
I'll do a differential
equation problem first.
00:01:39.970 --> 00:01:43.010
What's special about
symmetric matrices?
00:01:43.010 --> 00:01:46.060
Can we just say that in words?
00:01:46.060 --> 00:01:48.170
I'd better write
it down, though.
00:01:48.170 --> 00:01:50.960
What's special about
symmetric matrices?
00:01:50.960 --> 00:01:56.970
Their eigenvalues are real.
00:01:56.970 --> 00:02:00.970
The eigenvalues of a symmetric
matrix always come out real,
00:02:00.970 --> 00:02:04.800
and there always are
enough eigenvectors.
00:02:04.800 --> 00:02:06.930
Even if there are
repeated eigenvalues,
00:02:06.930 --> 00:02:09.910
there are enough
eigenvectors, and we
00:02:09.910 --> 00:02:13.180
can choose those eigenvectors
to be orthogonal.
00:02:13.180 --> 00:02:16.480
So if A equals A
transposed, the big fact
00:02:16.480 --> 00:02:22.620
will be that we
can diagonalize it,
00:02:22.620 --> 00:02:28.550
and those eigenvector
matrix, with the eigenvectors
00:02:28.550 --> 00:02:31.470
in the column, can be
an orthogonal matrix.
00:02:31.470 --> 00:02:36.600
So we get a Q
lambda Q transpose.
00:02:36.600 --> 00:02:43.010
That, in three symbols,
expresses a wonderful fact,
00:02:43.010 --> 00:02:46.250
a fundamental fact for
symmetric matrices.
00:02:46.250 --> 00:02:46.960
OK.
00:02:46.960 --> 00:02:49.740
Then, we went beyond
that fact to ask
00:02:49.740 --> 00:02:52.570
about positive
definite matrices, when
00:02:52.570 --> 00:02:54.370
the eigenvalues were positive.
00:02:54.370 --> 00:02:55.970
I'll do an example of that.
00:02:59.290 --> 00:03:01.580
Now we've left symmetry.
00:03:01.580 --> 00:03:05.000
Similar matrices are
any square matrices,
00:03:05.000 --> 00:03:10.730
but two matrices are similar
if they're related that way.
00:03:10.730 --> 00:03:14.590
And what's the key point
about similar matrices?
00:03:14.590 --> 00:03:16.770
Somehow, those matrices
are representing
00:03:16.770 --> 00:03:19.410
the same thing in
different basis,
00:03:19.410 --> 00:03:21.980
in chapter seven language.
00:03:21.980 --> 00:03:29.220
In chapter six language, what's
up with these similar matrices?
00:03:29.220 --> 00:03:31.860
What's the key fact,
the key positive fact
00:03:31.860 --> 00:03:33.570
about similar matrices?
00:03:33.570 --> 00:03:37.130
They have the same eigenvalues.
00:03:37.130 --> 00:03:39.070
Same eigenvalues.
00:03:39.070 --> 00:03:42.460
So if one of them grows,
the other one grows.
00:03:42.460 --> 00:03:50.380
If one of them decays to zero,
the other one decays to zero.
00:03:50.380 --> 00:03:53.050
Powers of A will look
like powers of B,
00:03:53.050 --> 00:03:55.440
because powers of
A and powers of B
00:03:55.440 --> 00:03:59.160
only differ by an M
inverse and an M way on the
00:03:59.160 --> 00:04:00.040
outside.
00:04:00.040 --> 00:04:05.250
So if these are similar,
then B to the k-th power
00:04:05.250 --> 00:04:09.160
is M inverse A to
the k-th power M.
00:04:09.160 --> 00:04:12.410
And that's why I
say, eh, this M, it
00:04:12.410 --> 00:04:14.480
does change the
eigenvectors, but it
00:04:14.480 --> 00:04:16.089
doesn't change the eigenvalues.
00:04:16.089 --> 00:04:21.209
So same lambdas.
00:04:21.209 --> 00:04:24.510
And then, finally, I've
got to review the point
00:04:24.510 --> 00:04:30.000
about the SVD, the Singular
Value Decomposition.
00:04:30.000 --> 00:04:30.500
OK.
00:04:30.500 --> 00:04:33.900
So that's what this
quiz has got to cover,
00:04:33.900 --> 00:04:37.720
and now I'll just take
problems from earlier exams,
00:04:37.720 --> 00:04:40.460
starting with a
differential equation.
00:04:40.460 --> 00:04:41.250
OK.
00:04:41.250 --> 00:04:43.360
And always ready for questions.
00:04:43.360 --> 00:04:46.640
So here is an exam
from about the year
00:04:46.640 --> 00:04:53.290
zero, and it has
a three by three.
00:04:53.290 --> 00:04:54.502
So that was --
00:04:57.280 --> 00:04:59.720
but it's a pretty
special-looking matrix,
00:04:59.720 --> 00:05:04.190
it's got zeroes on the diagonal,
it's got minus ones above,
00:05:04.190 --> 00:05:08.360
and it's got plus
ones like that.
00:05:08.360 --> 00:05:12.160
So that's the matrix A.
00:05:12.160 --> 00:05:12.880
OK.
00:05:12.880 --> 00:05:16.750
Step one is, well,
I want to solve that
00:05:16.750 --> 00:05:17.700
equation.
00:05:17.700 --> 00:05:19.970
I want to find the
general solution.
00:05:19.970 --> 00:05:22.790
I haven't given you
a u(0) here, so I'm
00:05:22.790 --> 00:05:24.900
looking for the
general solution,
00:05:24.900 --> 00:05:28.280
so now what's the form
of the general solution?
00:05:28.280 --> 00:05:30.520
With three arbitrary
constants going
00:05:30.520 --> 00:05:33.940
to be inside it, because
those will be used
00:05:33.940 --> 00:05:35.780
to match the initial condition.
00:05:35.780 --> 00:05:39.580
So the general
form is u at time t
00:05:39.580 --> 00:05:45.500
is some multiple of the
first special solution.
00:05:45.500 --> 00:05:50.080
The first special solution will
be growing like the eigenvalue,
00:05:50.080 --> 00:05:51.920
and it's the eigenvector.
00:05:51.920 --> 00:05:56.400
So that's a pure exponential
solution, just staying
00:05:56.400 --> 00:05:58.580
with that eigenvector.
00:05:58.580 --> 00:06:01.770
Of course, I haven't
found, yet, the eigenvalues
00:06:01.770 --> 00:06:02.580
and eigenvectors.
00:06:02.580 --> 00:06:04.900
That's, normally, the first job.
00:06:04.900 --> 00:06:08.880
Now, there will be second
one, growing like e
00:06:08.880 --> 00:06:13.160
to the lambda two, and a
third one growing like e
00:06:13.160 --> 00:06:16.450
to the lambda three.
00:06:16.450 --> 00:06:18.880
So we're all done --
00:06:18.880 --> 00:06:20.860
well, we haven't
done anything yet,
00:06:20.860 --> 00:06:22.100
actually.
00:06:22.100 --> 00:06:25.930
I've got to find the
eigenvalues and eigenvectors,
00:06:25.930 --> 00:06:30.720
and then I would match u(0)
by choosing the right three
00:06:30.720 --> 00:06:31.420
constants.
00:06:31.420 --> 00:06:31.950
OK.
00:06:31.950 --> 00:06:35.030
So now I ask -- ask you
about the eigenvalues
00:06:35.030 --> 00:06:38.790
and eigenvectors, and you look
at this matrix and what do you
00:06:38.790 --> 00:06:41.390
see in that matrix?
00:06:41.390 --> 00:06:47.740
Um, well, I guess we might
ask ourselves right away,
00:06:47.740 --> 00:06:50.760
is it singular?
00:06:50.760 --> 00:06:51.810
Is it singular?
00:06:51.810 --> 00:06:54.040
Because, if so, then we
really have a head start,
00:06:54.040 --> 00:06:56.810
we know one of the
eigenvalues is zero.
00:06:56.810 --> 00:06:57.990
Is that matrix singular?
00:07:01.660 --> 00:07:04.280
Eh, I don't know, do you
take the determinant to
00:07:04.280 --> 00:07:05.110
find out?
00:07:05.110 --> 00:07:08.530
Or maybe you look at the
first row and third row
00:07:08.530 --> 00:07:10.270
and say, hey, the
first row and third row
00:07:10.270 --> 00:07:15.180
are just opposite signs,
they're linear-dependent?
00:07:15.180 --> 00:07:17.930
The first column and third
column are dependent -- it's
00:07:17.930 --> 00:07:19.150
singular.
00:07:19.150 --> 00:07:22.610
So one eigenvalue is zero.
00:07:22.610 --> 00:07:24.520
Let's make that lambda one.
00:07:24.520 --> 00:07:26.150
Lambda one, then, will be zero.
00:07:26.150 --> 00:07:26.850
OK.
00:07:26.850 --> 00:07:30.000
Now we've got a couple of
other eigenvalues to find,
00:07:30.000 --> 00:07:33.430
and, I suppose the
simplest way is
00:07:33.430 --> 00:07:36.940
to look at A minus
lambda I So let
00:07:36.940 --> 00:07:43.720
me just put minus lambda
in here, minus ones above,
00:07:43.720 --> 00:07:45.520
ones below.
00:07:45.520 --> 00:07:51.430
But, actually, before
I do it, that matrix
00:07:51.430 --> 00:07:53.910
is not symmetric,
for sure, right?
00:07:53.910 --> 00:07:57.220
In fact, it's the very
opposite of symmetric.
00:07:57.220 --> 00:08:02.420
That matrix A transpose, how
is A transpose connected to A?
00:08:02.420 --> 00:08:04.020
It's negative A.
00:08:04.020 --> 00:08:07.890
It's an anti-symmetric
matrix, skew-symmetric matrix.
00:08:07.890 --> 00:08:11.230
And we've met, maybe,
a two-by-two example
00:08:11.230 --> 00:08:14.370
of skew-symmetric
matrices, and let
00:08:14.370 --> 00:08:17.687
me just say, what's the
deal with their eigenvalues?
00:08:17.687 --> 00:08:18.645
They're pure imaginary.
00:08:21.400 --> 00:08:23.440
They'll be on the
imaginary axis,
00:08:23.440 --> 00:08:26.710
there be some
multiple of I if it's
00:08:26.710 --> 00:08:29.090
an anti-symmetric,
skew-symmetric matrix.
00:08:29.090 --> 00:08:31.360
So I'm looking for
multiples of I,
00:08:31.360 --> 00:08:33.730
and of course,
that's zero times I,
00:08:33.730 --> 00:08:37.760
that's on the imaginary axis,
but maybe I just do it out,
00:08:37.760 --> 00:08:38.350
here.
00:08:38.350 --> 00:08:39.740
Lambda cubed.
00:08:39.740 --> 00:08:42.600
well, maybe that's
minus lambda cubed,
00:08:42.600 --> 00:08:44.480
and then a zero and a zero.
00:08:44.480 --> 00:08:47.870
Zero, and then maybe I
have a plus a lambda,
00:08:47.870 --> 00:08:52.610
and another plus lambda, but
those go with a minus sign.
00:08:52.610 --> 00:08:55.080
Am I getting minus two
lambda equals zero?
00:08:58.640 --> 00:08:59.940
So.
00:08:59.940 --> 00:09:04.810
So I'm solving lambda cube
plus two lambda equals zero.
00:09:04.810 --> 00:09:09.070
So one root factors out
lambda, and the the rest
00:09:09.070 --> 00:09:11.571
is lambda squared plus two.
00:09:11.571 --> 00:09:12.070
OK.
00:09:12.070 --> 00:09:14.310
This is going the
way we expect, right?
00:09:14.310 --> 00:09:22.360
Because this gives the root
lambda equals zero, and gives
00:09:22.360 --> 00:09:27.630
the other two roots, which
are lambda equal what?
00:09:27.630 --> 00:09:30.200
The solutions of when
is lambda squared
00:09:30.200 --> 00:09:36.510
plus two equals zero then the
eigenvalues those guys, what
00:09:36.510 --> 00:09:38.610
are they?
00:09:38.610 --> 00:09:41.260
They're a multiple of i,
they're just square root of two
00:09:41.260 --> 00:09:42.150
i.
00:09:42.150 --> 00:09:45.010
When I set this
equals to zero, I
00:09:45.010 --> 00:09:48.800
have lambda squared equal
to minus two, right?
00:09:48.800 --> 00:09:50.710
To make that zero?
00:09:50.710 --> 00:09:54.180
And the roots are
square root of two i
00:09:54.180 --> 00:09:58.060
and minus the square
root of two i.
00:09:58.060 --> 00:09:59.540
So now I know what those are.
00:09:59.540 --> 00:10:01.070
I'll put those in, now.
00:10:01.070 --> 00:10:03.840
Either the zero t is just a one.
00:10:03.840 --> 00:10:06.840
That's just a one.
00:10:06.840 --> 00:10:13.700
This is square root of
two I and this is minus
00:10:13.700 --> 00:10:18.280
square root of two I.
00:10:18.280 --> 00:10:22.590
So, is the solution
decaying to zero?
00:10:22.590 --> 00:10:25.210
Is this a completely
stable problem
00:10:25.210 --> 00:10:28.630
where the solution
is going to zero?
00:10:28.630 --> 00:10:31.200
No.
00:10:31.200 --> 00:10:34.620
In fact, all these things
are staying the same size.
00:10:34.620 --> 00:10:39.290
This thing is getting
multiplied by this number.
00:10:39.290 --> 00:10:46.880
e to the I something t, that's
a number that has magnitude one,
00:10:46.880 --> 00:10:50.160
and sort of wanders
around the unit circle.
00:10:50.160 --> 00:10:51.660
Same for this.
00:10:51.660 --> 00:10:55.900
So that the solution doesn't
blow up, and it doesn't go to
00:10:55.900 --> 00:10:56.520
zero.
00:10:56.520 --> 00:10:57.410
OK.
00:10:57.410 --> 00:10:59.670
And to find out
what it actually is,
00:10:59.670 --> 00:11:02.370
we would have to plug
in initial conditions.
00:11:02.370 --> 00:11:04.410
But actually, the
next question I ask
00:11:04.410 --> 00:11:12.280
is, when does the solution
return to its initial value?
00:11:12.280 --> 00:11:15.910
I won't even say what's
the initial value.
00:11:15.910 --> 00:11:22.540
This is a case in which
I think this solution is
00:11:22.540 --> 00:11:25.350
periodic after.
00:11:25.350 --> 00:11:31.820
At t equals zero, it
starts with c1, c2, and c3,
00:11:31.820 --> 00:11:36.500
and then at some value of
t, it comes back to that.
00:11:36.500 --> 00:11:38.440
So that's a very
special question,
00:11:38.440 --> 00:11:40.280
Well, let's just
take three seconds,
00:11:40.280 --> 00:11:44.040
because that special question
isn't likely to be on the quiz.
00:11:44.040 --> 00:11:49.910
But it comes back
to the start, when?
00:11:49.910 --> 00:11:55.230
Well, whenever we have e to
the two pi i, that's one,
00:11:55.230 --> 00:11:56.450
and we've come back again.
00:11:56.450 --> 00:11:58.700
So it comes back to the start.
00:11:58.700 --> 00:12:07.740
It's periodic, when this
square root of two i --
00:12:07.740 --> 00:12:10.930
shall I call it capital
T, for the period?
00:12:10.930 --> 00:12:17.270
For that particular T, if
that equals two pi i, then e
00:12:17.270 --> 00:12:21.090
to this thing is one, and
we've come around again.
00:12:21.090 --> 00:12:26.250
So the period is T is
determined here, cancel the i-s,
00:12:26.250 --> 00:12:30.970
and T is pi times the
square root of two.
00:12:30.970 --> 00:12:32.270
So that's pretty neat.
00:12:32.270 --> 00:12:35.610
We get all the information
about all solutions,
00:12:35.610 --> 00:12:39.230
we haven't fixed on only
one particular solution,
00:12:39.230 --> 00:12:41.450
but it comes around again.
00:12:41.450 --> 00:12:43.730
So this was probably
my first chance
00:12:43.730 --> 00:12:46.270
to say something
about the whole family
00:12:46.270 --> 00:12:49.760
of anti-symmetric,
skew-symmetric matrices.
00:12:49.760 --> 00:12:50.380
OK.
00:12:50.380 --> 00:12:56.880
And then, finally, I asked,
take two eigenvectors (again,
00:12:56.880 --> 00:12:59.450
I haven't computed
the eigenvectors)
00:12:59.450 --> 00:13:02.690
and it turns out
they're orthogonal.
00:13:02.690 --> 00:13:03.500
They're orthogonal.
00:13:03.500 --> 00:13:06.500
The eigenvectors of
a symmetric matrix,
00:13:06.500 --> 00:13:10.250
or a skew-symmetric matrix,
are always orthogonal.
00:13:13.200 --> 00:13:18.020
I guess may conscience
makes me tell you,
00:13:18.020 --> 00:13:23.950
what are all the matrices that
have orthogonal eigenvectors?
00:13:23.950 --> 00:13:27.040
And symmetric is the
most important class,
00:13:27.040 --> 00:13:28.640
so that's the one
we've spoken about.
00:13:28.640 --> 00:13:32.800
But let me just put that
little fact down, here.
00:13:32.800 --> 00:13:36.661
Orthogonal x-s.
00:13:36.661 --> 00:13:37.202
eigenvectors.
00:13:41.430 --> 00:13:44.200
A matrix has orthogonal
eigenvectors,
00:13:44.200 --> 00:13:47.430
the exact condition -- it's
quite beautiful that I can tell
00:13:47.430 --> 00:13:49.340
you exactly when that happens.
00:13:49.340 --> 00:13:55.290
It happens when A times A
transpose equals A transpose
00:13:55.290 --> 00:14:00.290
times A. Any time
that's the condition
00:14:00.290 --> 00:14:03.700
for orthogonal eigenvectors.
00:14:03.700 --> 00:14:09.040
And because we're interested
in special families of vectors,
00:14:09.040 --> 00:14:12.690
tell me some special
families that fit.
00:14:12.690 --> 00:14:15.410
This is the whole requirement.
00:14:15.410 --> 00:14:21.120
That's a pretty special
requirement most matrices have.
00:14:21.120 --> 00:14:23.070
So the average
three-by-three matrix
00:14:23.070 --> 00:14:26.240
has three eigenvectors,
but not orthogonal.
00:14:26.240 --> 00:14:29.380
But if it happens to
commute with its transpose,
00:14:29.380 --> 00:14:34.010
then, wonderfully, the
eigenvectors are orthogonal.
00:14:34.010 --> 00:14:39.250
Now, do you see how symmetric
matrices pass this test?
00:14:39.250 --> 00:14:40.340
Of course.
00:14:40.340 --> 00:14:43.790
If A transpose equals A, then
both sides are A squared,
00:14:43.790 --> 00:14:45.640
we've got it.
00:14:45.640 --> 00:14:49.410
How do anti-symmetric
matrices pass this test?
00:14:49.410 --> 00:14:54.060
If A transpose equals
minus A, then we've
00:14:54.060 --> 00:14:58.220
got it again, because we've got
minus A squared on both sides.
00:14:58.220 --> 00:15:00.050
So that's another group.
00:15:00.050 --> 00:15:03.090
And finally, let me ask you
about our other favorite
00:15:03.090 --> 00:15:06.950
family, orthogonal matrices.
00:15:06.950 --> 00:15:11.730
Do orthogonal matrices pass
this test, if A is a Q,
00:15:11.730 --> 00:15:14.990
do they pass the test for
orthogonal eigenvectors.
00:15:14.990 --> 00:15:22.320
Well, if A is Q, an orthogonal
matrix, what is Q transpose Q?
00:15:22.320 --> 00:15:23.020
It's I.
00:15:23.020 --> 00:15:25.310
And what is Q Q transpose?
00:15:25.310 --> 00:15:28.010
It's I, we're talking
square matrices here.
00:15:28.010 --> 00:15:30.100
So yes, it passes the test.
00:15:30.100 --> 00:15:36.600
So the special cases are
symmetric, anti-symmetric
00:15:36.600 --> 00:15:41.080
(I'll say skew-symmetric,)
and orthogonal.
00:15:41.080 --> 00:15:44.280
Those are the three
important special classes
00:15:44.280 --> 00:15:45.710
that are in this family.
00:15:45.710 --> 00:15:46.210
OK.
00:15:46.210 --> 00:15:52.860
That's like a comment that,
could have been made back in,
00:15:52.860 --> 00:15:54.870
section six point four.
00:15:54.870 --> 00:16:04.390
OK, I can pursue the
differential equations, also
00:16:04.390 --> 00:16:09.090
this question, didn't
ask you to tell me,
00:16:09.090 --> 00:16:13.920
how would I find this matrix
exponential, e to the At?
00:16:13.920 --> 00:16:15.050
So can I erase this?
00:16:15.050 --> 00:16:17.050
I'll just stay with this same...
00:16:19.690 --> 00:16:23.770
how would I find e to the At?
00:16:23.770 --> 00:16:27.010
Because, how does that come in?
00:16:27.010 --> 00:16:30.140
That's the key matrix for
a differential equation,
00:16:30.140 --> 00:16:32.460
because the solution is --
00:16:32.460 --> 00:16:38.520
the solution is
u(t) is e^(At) u(0).
00:16:38.520 --> 00:16:42.060
So this is like the
fundamental matrix
00:16:42.060 --> 00:16:48.400
that multiplies the given
function and gives the answer.
00:16:48.400 --> 00:16:53.630
And how would we compute
it if we wanted that?
00:16:53.630 --> 00:16:56.830
We don't always have to
find e to the At, because I
00:16:56.830 --> 00:17:00.500
can go directly to the answer
without any e to the At-s,
00:17:00.500 --> 00:17:06.440
but hiding here is an e to the
At, and how would I compute it?
00:17:06.440 --> 00:17:10.026
Well, if A is diagonalizable.
00:17:12.589 --> 00:17:21.510
So I'm now going to put in my
usual if A can be diagonalized
00:17:21.510 --> 00:17:25.240
(and everybody remember
that there is an if there,
00:17:25.240 --> 00:17:28.820
because it might not
have enough eigenvectors)
00:17:28.820 --> 00:17:33.330
this example does have enough,
random matrices have enough.
00:17:33.330 --> 00:17:36.720
So if we can diagonalize, then
we get a nice formula for this,
00:17:36.720 --> 00:17:40.050
because an S comes way
out at the beginning,
00:17:40.050 --> 00:17:42.740
and S inverse comes
way out at the end,
00:17:42.740 --> 00:17:47.450
and we only have to take
the exponential of lambda.
00:17:47.450 --> 00:17:49.980
And that's just a
diagonal matrix,
00:17:49.980 --> 00:17:53.780
so that's just e
the lambda one t,
00:17:53.780 --> 00:18:00.030
these guys are showing up,
now, in e to the lambda nt.
00:18:00.030 --> 00:18:01.070
OK?
00:18:01.070 --> 00:18:03.510
That's a really quick
review of that formula.
00:18:06.040 --> 00:18:08.780
It's something we can
compute it quickly
00:18:08.780 --> 00:18:11.250
if we have done the
S and lambda part.
00:18:13.770 --> 00:18:15.470
If we know S and
lambda, then it's
00:18:15.470 --> 00:18:17.440
not hard to take that step.
00:18:17.440 --> 00:18:20.770
OK, that's some comments
on differential equations.
00:18:20.770 --> 00:18:28.330
I would like to go on to a next
question that I started here.
00:18:28.330 --> 00:18:33.410
And it's, got several parts,
and I can just read it out.
00:18:33.410 --> 00:18:37.320
What we're given is a
three-by-three matrix,
00:18:37.320 --> 00:18:41.900
and we're told its eigenvalues,
except one of these
00:18:41.900 --> 00:18:47.700
is, like, we don't know, and
we're told the eigenvectors.
00:18:47.700 --> 00:18:50.500
And I want to ask
you about the matrix.
00:18:50.500 --> 00:18:51.190
OK.
00:18:51.190 --> 00:18:54.530
So, first question.
00:18:54.530 --> 00:18:56.430
Is the matrix diagonalizable?
00:18:59.000 --> 00:19:03.030
And I really mean for
which c, because I
00:19:03.030 --> 00:19:06.850
don't know c, so my
questions will all be,
00:19:06.850 --> 00:19:12.340
for which is there a condition
on c, does one c work.
00:19:12.340 --> 00:19:17.270
But your answer should tell
me all the c-s that work.
00:19:17.270 --> 00:19:21.390
I'm not asking for you to
tell me, well, c equal four,
00:19:21.390 --> 00:19:22.660
yes, that checks out.
00:19:22.660 --> 00:19:27.927
I want to know all the c-s
that make it diagonalizable.
00:19:34.950 --> 00:19:36.640
OK?
00:19:36.640 --> 00:19:39.440
What's the real
on diagonalizable?
00:19:39.440 --> 00:19:42.212
We need enough
eigenvectors, right?
00:19:42.212 --> 00:19:43.920
We don't care what
those eigenvalues are,
00:19:43.920 --> 00:19:46.710
it's eigenvectors that
count for diagonalizable,
00:19:46.710 --> 00:19:49.220
and we need three
independent ones,
00:19:49.220 --> 00:19:52.340
and are those three
guys independent?
00:19:52.340 --> 00:19:53.580
Yes.
00:19:53.580 --> 00:19:56.300
Actually, let's look
at them for a moment.
00:19:56.300 --> 00:19:59.921
What do you see about those
three vectors right away?
00:19:59.921 --> 00:20:01.170
They're more than independent.
00:20:04.380 --> 00:20:09.730
Can you see why those
three got chosen?
00:20:09.730 --> 00:20:15.780
Because it will come up in the
next part, they're orthogonal.
00:20:15.780 --> 00:20:17.920
Those eigenvectors
are orthogonal.
00:20:17.920 --> 00:20:19.710
They're certainly independent.
00:20:19.710 --> 00:20:28.220
So the answer to diagonalizable
is, yes, all c, all c.
00:20:28.220 --> 00:20:30.760
Doesn't matter. c could
be a repeated guy,
00:20:30.760 --> 00:20:32.360
but we've got
enough eigenvectors,
00:20:32.360 --> 00:20:33.930
so that's what we care about.
00:20:33.930 --> 00:20:36.400
OK, second question.
00:20:36.400 --> 00:20:38.365
For which values of
c is it symmetric?
00:20:40.960 --> 00:20:46.000
OK, what's the
answer to that one?
00:20:48.650 --> 00:20:53.420
If we know the same setup if
we know that much about it,
00:20:53.420 --> 00:20:55.330
we know those
eigenvectors, and we've
00:20:55.330 --> 00:21:02.850
noticed they're orthogonal,
then which c-s will work?
00:21:02.850 --> 00:21:07.800
So the eigenvalues of that
symmetric matrix have to be
00:21:07.800 --> 00:21:08.490
real.
00:21:08.490 --> 00:21:11.780
So all real c.
00:21:11.780 --> 00:21:17.300
If c was i, the matrix
wouldn't have been symmetric.
00:21:17.300 --> 00:21:24.040
But if c is a real number, then
we've got real eigenvalues,
00:21:24.040 --> 00:21:25.920
we've got orthogonal
eigenvectors,
00:21:25.920 --> 00:21:27.400
that matrix is symmetric.
00:21:27.400 --> 00:21:28.790
OK, positive definite.
00:21:28.790 --> 00:21:40.630
OK, now this is a
sub-case of symmetric,
00:21:40.630 --> 00:21:45.900
so we need c to be real, so
we've got a symmetric matrix,
00:21:45.900 --> 00:21:50.360
but we also want the thing
to be positive definite.
00:21:50.360 --> 00:21:52.340
Now, we're looking
at eigenvalues,
00:21:52.340 --> 00:21:54.740
we've got a lot of tests
for positive definite,
00:21:54.740 --> 00:21:57.250
but eigenvalues,
if we know them,
00:21:57.250 --> 00:22:01.100
is certainly a good,
quick, clean test.
00:22:01.100 --> 00:22:05.570
Could this matrix be
positive definite?
00:22:05.570 --> 00:22:06.640
No.
00:22:06.640 --> 00:22:10.180
No, because it's got
an eigenvalue zero.
00:22:10.180 --> 00:22:12.640
It could be positive
semi-definite,
00:22:12.640 --> 00:22:15.900
you know, like
consolation prize,
00:22:15.900 --> 00:22:19.320
if c was greater
or equal to zero,
00:22:19.320 --> 00:22:21.680
it would be positive
semi-definite.
00:22:21.680 --> 00:22:25.520
But it's not, no.
00:22:25.520 --> 00:22:30.670
Semi-definite, if I put that
comment in, semi-definite,
00:22:30.670 --> 00:22:35.010
that the condition would be
c greater or equal to zero.
00:22:35.010 --> 00:22:36.300
That would be all right.
00:22:36.300 --> 00:22:37.220
OK.
00:22:37.220 --> 00:22:38.330
Next part.
00:22:38.330 --> 00:22:39.675
Is it a Markov matrix?
00:22:44.260 --> 00:22:44.870
Hm.
00:22:44.870 --> 00:22:50.440
Could this matrix be, if I
choose the number c correctly,
00:22:50.440 --> 00:22:52.040
a Markov matrix?
00:22:58.750 --> 00:23:02.700
Well, what do we know
about Markov matrices?
00:23:02.700 --> 00:23:05.320
Mainly, we know something
about their eigenvalues.
00:23:05.320 --> 00:23:10.500
One eigenvalue is always one,
and the other eigenvalues
00:23:10.500 --> 00:23:13.430
are smaller.
00:23:13.430 --> 00:23:14.690
Not larger.
00:23:14.690 --> 00:23:17.380
So an eigenvalue
two can't happen.
00:23:17.380 --> 00:23:22.000
So the answer is, no, not a ma-
that's never a Markov matrix.
00:23:22.000 --> 00:23:22.750
OK?
00:23:22.750 --> 00:23:29.970
And finally, could one half
of A be a projection matrix?
00:23:29.970 --> 00:23:32.820
So could it- could this --
eh-eh could this be twice
00:23:32.820 --> 00:23:33.890
a projection matrix?
00:23:33.890 --> 00:23:35.840
So let me write it this way.
00:23:35.840 --> 00:23:39.530
Could A over two be
a projection matrix?
00:23:44.820 --> 00:23:46.820
OK, what are
projection matrices?
00:23:46.820 --> 00:23:48.650
They're real.
00:23:48.650 --> 00:23:53.160
I mean, th- they're symmetric,
so their eigenvalues are real.
00:23:53.160 --> 00:23:56.680
But more than that, we know what
those eigenvalues have to be.
00:23:56.680 --> 00:24:01.150
What do the eigenvalues of a
projection matrix have to be?
00:24:01.150 --> 00:24:05.670
See, that any nice
matrix we've got
00:24:05.670 --> 00:24:08.490
an idea about its eigenvalues.
00:24:08.490 --> 00:24:12.980
So the eigenvalues of
projection matrices are zero and
00:24:12.980 --> 00:24:13.970
one.
00:24:13.970 --> 00:24:16.710
Zero and one, only.
00:24:16.710 --> 00:24:21.510
Because P squared equals P,
let me call this matrix P,
00:24:21.510 --> 00:24:26.510
so P squared equals P, so
lambda squared equals lambda,
00:24:26.510 --> 00:24:30.570
because eigenvalues of P
squared are lambda squared,
00:24:30.570 --> 00:24:37.520
and we must have that, so
lambda equals zero or one.
00:24:37.520 --> 00:24:38.140
OK.
00:24:38.140 --> 00:24:42.060
Now what value of
c will work there?
00:24:42.060 --> 00:24:48.250
So, then, there are some
value that will work,
00:24:48.250 --> 00:24:50.300
and what will work?
00:24:50.300 --> 00:24:56.340
c equals zero will work,
or what else will work?
00:24:59.310 --> 00:25:02.690
c equal to two.
00:25:02.690 --> 00:25:06.260
Because if c is two, then
when we divide by two,
00:25:06.260 --> 00:25:09.880
this Eigenvalue of
two will drop to one,
00:25:09.880 --> 00:25:13.110
and so will the other one,
so, or c equal to two.
00:25:13.110 --> 00:25:15.640
OK, those are the
guys that will work,
00:25:15.640 --> 00:25:21.110
and it was the fact that those
eigenvectors were orthogonal,
00:25:21.110 --> 00:25:23.780
the fact that those
eigenvectors were orthogonal
00:25:23.780 --> 00:25:26.170
carried us a lot
of the way, here.
00:25:26.170 --> 00:25:29.400
If they weren't orthogonal, then
symmetric would have been dead,
00:25:29.400 --> 00:25:31.420
positive definite
would have been dead,
00:25:31.420 --> 00:25:33.150
projection would have been dead.
00:25:33.150 --> 00:25:37.090
But those eigenvectors
were orthogonal,
00:25:37.090 --> 00:25:40.180
so it came down to
the eigenvalues.
00:25:40.180 --> 00:25:45.270
OK, that was like a chance to
review a lot of this chapter.
00:25:50.790 --> 00:25:56.030
Shall I jump to the singular
value decomposition,
00:25:56.030 --> 00:26:04.140
then, as the third, topic
for, for the review?
00:26:04.140 --> 00:26:06.080
OK, so I'm going
to. jump to this.
00:26:06.080 --> 00:26:06.580
OK.
00:26:13.950 --> 00:26:16.990
So this is the singular
value decomposition,
00:26:16.990 --> 00:26:21.070
known to everybody as the SVD.
00:26:21.070 --> 00:26:27.720
And that's a factorization
of A into orthogonal times
00:26:27.720 --> 00:26:33.830
diagonal times orthogonal.
00:26:33.830 --> 00:26:41.030
And we always call those U
and sigma and V transpose.
00:26:41.030 --> 00:26:42.300
OK.
00:26:42.300 --> 00:26:46.660
And the key to that --
00:26:46.660 --> 00:26:51.170
this is for every
matrix, every A, every A.
00:26:51.170 --> 00:26:54.110
Rectangular, doesn't
matter, whatever,
00:26:54.110 --> 00:26:56.740
has this decomposition.
00:26:56.740 --> 00:26:59.070
So it's really important.
00:26:59.070 --> 00:27:04.930
And the key to it is to look
at things like A transpose A.
00:27:04.930 --> 00:27:07.360
Can we remember what
happens with A transpose A?
00:27:07.360 --> 00:27:11.300
If I just transpose that
I get V sigma transpose U
00:27:11.300 --> 00:27:15.420
transpose, that's
multiplying A, which is U,
00:27:15.420 --> 00:27:24.850
sigma V transpose, and the
result is V on the outside,
00:27:24.850 --> 00:27:27.990
s- U transpose U
is the identity,
00:27:27.990 --> 00:27:30.930
because it's an
orthogonal matrix.
00:27:30.930 --> 00:27:34.550
So I'm just left with
sigma transpose sigma
00:27:34.550 --> 00:27:39.340
in the middle, that's
a diagonal, possibly
00:27:39.340 --> 00:27:42.960
rectangular diagonal by its
transpose, so the result,
00:27:42.960 --> 00:27:46.036
this is orthogonal,
diagonal, orthogonal.
00:27:49.840 --> 00:27:55.620
So, I guess, actually, this
is the SVD for A transpose A.
00:27:55.620 --> 00:27:59.930
Here I see orthogonal,
diagonal, and orthogonal.
00:27:59.930 --> 00:28:00.440
Great.
00:28:00.440 --> 00:28:07.000
But a little more is happening.
00:28:07.000 --> 00:28:09.580
For A transpose
A, the difference
00:28:09.580 --> 00:28:13.690
is, the orthogonal
guys are the same.
00:28:13.690 --> 00:28:15.640
It's V and V transpose.
00:28:15.640 --> 00:28:17.290
What I seeing here?
00:28:17.290 --> 00:28:21.950
I'm seeing the factorization
for a symmetric matrix.
00:28:21.950 --> 00:28:23.220
This thing is symmetric.
00:28:26.790 --> 00:28:30.590
So in a symmetric case,
U is the same as V.
00:28:30.590 --> 00:28:33.210
U is the same as V for
this symmetric matrix,
00:28:33.210 --> 00:28:34.990
and, of course, we
see it happening.
00:28:34.990 --> 00:28:35.540
OK.
00:28:35.540 --> 00:28:39.760
So that tells us,
right away, what V is.
00:28:39.760 --> 00:28:49.650
V is the eigenvector
matrix for A transpose A.
00:28:49.650 --> 00:28:50.490
OK.
00:28:50.490 --> 00:28:57.070
Now, if you were here when I
lectured about this topic, when
00:28:57.070 --> 00:29:00.600
I gave the topic on singular
value decompositions,
00:29:00.600 --> 00:29:03.180
you'll remember that
I got into trouble.
00:29:06.150 --> 00:29:09.860
I'm sorry to remember that
myself, but it happened.
00:29:09.860 --> 00:29:10.550
OK.
00:29:10.550 --> 00:29:13.680
How did it happen?
00:29:13.680 --> 00:29:16.900
I was in great shape for
a while, cruising along.
00:29:16.900 --> 00:29:20.120
So I found the eigenvectors
for A transpose A.
00:29:20.120 --> 00:29:21.870
Good.
00:29:21.870 --> 00:29:24.800
I found the singular
values, what were they?
00:29:24.800 --> 00:29:26.520
What were the singular values?
00:29:26.520 --> 00:29:32.720
The singular value
number i, or --
00:29:32.720 --> 00:29:36.890
these are the guys in sigma --
00:29:36.890 --> 00:29:39.770
this is diagonal with
the number sigma in it.
00:29:39.770 --> 00:29:42.910
This diagonal is
sigma one, sigma two,
00:29:42.910 --> 00:29:46.090
up to the rank, sigma r,
those are the non-zero ones.
00:29:48.830 --> 00:29:51.100
So I found those,
and what are they?
00:29:51.100 --> 00:29:53.130
Remind me about that?
00:29:53.130 --> 00:29:59.630
Well, here, I'm seeing them
squared, so their squares are
00:29:59.630 --> 00:30:03.470
the eigenvalues
of A transpose A.
00:30:03.470 --> 00:30:04.760
Good.
00:30:04.760 --> 00:30:09.160
So I just take the square root,
if I want the eigenvalues of A
00:30:09.160 --> 00:30:10.020
transpose --
00:30:10.020 --> 00:30:11.990
If I want the sigmas
and I know these,
00:30:11.990 --> 00:30:14.540
I take the square root,
the positive square root.
00:30:14.540 --> 00:30:16.730
OK.
00:30:16.730 --> 00:30:20.610
Where did I run into trouble?
00:30:20.610 --> 00:30:25.220
Well, then, my final
step was to find U.
00:30:25.220 --> 00:30:28.270
And I didn't read the book.
00:30:28.270 --> 00:30:35.480
So, I did something that was
practically right, but --
00:30:35.480 --> 00:30:38.880
well, I guess practically
right is not quite the same.
00:30:38.880 --> 00:30:44.650
OK, so I thought, OK, I'll
look at A A transpose.
00:30:44.650 --> 00:30:47.420
What happened when I
looked at A A transpose?
00:30:47.420 --> 00:30:51.070
Let me just put it here,
and then I can feel it.
00:30:51.070 --> 00:30:53.620
OK, so here's A A transpose.
00:30:57.120 --> 00:31:01.050
So that's U sigma V
transpose, that's A,
00:31:01.050 --> 00:31:05.240
and then the transpose
is V sigma transpose,
00:31:05.240 --> 00:31:06.300
U sigma transpose.
00:31:06.300 --> 00:31:07.930
Fine.
00:31:07.930 --> 00:31:10.610
And then, in the middle
is the identity again,
00:31:10.610 --> 00:31:12.570
so it looks great.
00:31:12.570 --> 00:31:17.050
U sigma sigma
transpose, U transpose.
00:31:17.050 --> 00:31:18.760
Fine.
00:31:18.760 --> 00:31:26.570
All good, and now
these columns of U
00:31:26.570 --> 00:31:29.900
are the eigenvectors,
that's U is the eigenvector
00:31:29.900 --> 00:31:33.120
matrix for this guy.
00:31:33.120 --> 00:31:36.960
That was correct,
so I did that fine.
00:31:36.960 --> 00:31:38.600
Where did something go wrong?
00:31:38.600 --> 00:31:40.820
A sign went wrong.
00:31:40.820 --> 00:31:44.570
A sign went wrong because --
and now -- now I see, actually,
00:31:44.570 --> 00:31:49.140
somebody told me
right after class,
00:31:49.140 --> 00:31:53.910
we can't tell from this
description which sign to give
00:31:53.910 --> 00:31:55.200
the eigenvectors.
00:31:55.200 --> 00:32:00.570
If these are the
eigenvectors of this matrix,
00:32:00.570 --> 00:32:02.660
well, if you give
me an eigenvector
00:32:02.660 --> 00:32:04.790
and I change all
its signs, we've
00:32:04.790 --> 00:32:06.920
still got another eigenvector.
00:32:06.920 --> 00:32:08.970
So what I wasn't
able to determine
00:32:08.970 --> 00:32:13.940
(and I had a fifty-fifty
change and life let me down,)
00:32:13.940 --> 00:32:16.600
the signs I just
happened to pick
00:32:16.600 --> 00:32:19.070
for the eigenvectors,
one of them
00:32:19.070 --> 00:32:21.750
I should have reversed the sign.
00:32:21.750 --> 00:32:27.190
So, from this, I can't tell
whether the eigenvector
00:32:27.190 --> 00:32:31.150
or its negative is the
right one to use in there.
00:32:31.150 --> 00:32:34.750
So the right way to
do it is to, having
00:32:34.750 --> 00:32:38.640
settled on the
signs, the Vs also, I
00:32:38.640 --> 00:32:42.100
don't know which sign to
choose, but I choose one.
00:32:42.100 --> 00:32:43.220
I choose one.
00:32:43.220 --> 00:32:50.290
And then, instead,
I should have used
00:32:50.290 --> 00:32:53.950
the one that tells me what
sign to choose, the rule
00:32:53.950 --> 00:33:02.330
that A times a V is
sigma times the U.
00:33:02.330 --> 00:33:07.140
So, having decided on
the V, I multiply by A,
00:33:07.140 --> 00:33:09.640
I'll notice the factor
sigma coming out,
00:33:09.640 --> 00:33:11.520
and there will be a
unit vector there,
00:33:11.520 --> 00:33:17.310
and I now know
exactly what it is,
00:33:17.310 --> 00:33:20.380
and not only up to
a change of sign.
00:33:20.380 --> 00:33:22.390
So that's the good
and, of course,
00:33:22.390 --> 00:33:25.910
this is the main
point about the SVD.
00:33:25.910 --> 00:33:28.210
That's the point that
we've diagonalized,
00:33:28.210 --> 00:33:32.950
that's A times the
matrix of Vs equals
00:33:32.950 --> 00:33:37.710
U times the diagonal
matrix of sigmas.
00:33:37.710 --> 00:33:39.470
That's the same as that.
00:33:39.470 --> 00:33:39.970
OK.
00:33:39.970 --> 00:33:47.800
So that's, like,
correcting the wrong sign
00:33:47.800 --> 00:33:50.000
from that earlier lecture.
00:33:50.000 --> 00:33:52.810
And that would complete that,
so that's how you would compute
00:33:52.810 --> 00:33:54.040
the SVD.
00:33:54.040 --> 00:33:58.380
Now, on the quiz, I going to
ask -- well, maybe on the final.
00:33:58.380 --> 00:34:01.010
So we've got quiz
and final ahead.
00:34:01.010 --> 00:34:05.400
Sometimes, you might be asked
to find the SVD if I give you
00:34:05.400 --> 00:34:10.870
the matrix -- let me come
back, now, to the main board --
00:34:10.870 --> 00:34:17.880
or, I might give you the pieces.
00:34:17.880 --> 00:34:21.810
And I might ask you
something about the matrix.
00:34:21.810 --> 00:34:31.580
For example, suppose I
ask you, oh, let's say,
00:34:31.580 --> 00:34:36.590
if I tell you what sigma is --
00:34:36.590 --> 00:34:37.460
OK.
00:34:37.460 --> 00:34:39.230
Let's take one example.
00:34:39.230 --> 00:34:43.820
Suppose sigma is --
00:34:43.820 --> 00:34:46.350
so all that's how we
would compute them.
00:34:46.350 --> 00:34:48.070
But now, suppose
I give you these.
00:34:48.070 --> 00:34:52.320
Suppose I give you sigma
is, say, three two.
00:34:57.130 --> 00:35:02.110
And I tell you that U
has a couple of columns,
00:35:02.110 --> 00:35:04.580
and V has a couple of columns.
00:35:07.910 --> 00:35:10.330
OK.
00:35:10.330 --> 00:35:12.600
Those are orthogonal
columns, of course,
00:35:12.600 --> 00:35:14.630
because U and V are orthogonal.
00:35:14.630 --> 00:35:16.120
I'm just sort of,
like, getting you
00:35:16.120 --> 00:35:19.440
to think about the SVD,
because we only had that one
00:35:19.440 --> 00:35:22.570
lecture about it,
and one homework,
00:35:22.570 --> 00:35:28.460
and, what kind of a
matrix have I got here?
00:35:28.460 --> 00:35:31.970
What do I know
about this matrix?
00:35:31.970 --> 00:35:35.540
All I really know right now
is that its singular values,
00:35:35.540 --> 00:35:39.390
those sigmas are three and
two, and the only thing
00:35:39.390 --> 00:35:43.190
interesting that I can see in
that is that they're not zero.
00:35:43.190 --> 00:35:48.390
I know that this matrix
is non-singular, right?
00:35:48.390 --> 00:35:51.570
That's invertible, I don't
have any zero eigenvalues,
00:35:51.570 --> 00:35:54.710
and zero singular values,
that's invertible,
00:35:54.710 --> 00:36:02.290
there's a typical SVD for a
nice two-by-two non-singular
00:36:02.290 --> 00:36:04.990
invertible good matrix.
00:36:04.990 --> 00:36:07.480
If I actually gave you
a matrix, then you'd
00:36:07.480 --> 00:36:10.390
have to find the Us and
the Vs as we just spoke.
00:36:10.390 --> 00:36:12.000
But, there.
00:36:12.000 --> 00:36:16.400
Now, what if the two
wasn't a two but it was --
00:36:16.400 --> 00:36:18.520
well, let me make an
extreme case, here --
00:36:18.520 --> 00:36:20.050
suppose it was minus five.
00:36:23.220 --> 00:36:24.810
That's wrong, right away.
00:36:24.810 --> 00:36:28.590
That's not a singular
value decomposition, right?
00:36:28.590 --> 00:36:30.840
The singular values
are not negative.
00:36:30.840 --> 00:36:36.011
So that's not a singular value
decomposition, and forget it.
00:36:36.011 --> 00:36:36.510
OK.
00:36:36.510 --> 00:36:40.200
So let me ask you
about that one.
00:36:40.200 --> 00:36:42.095
What can you tell me
about that matrix?
00:36:45.090 --> 00:36:47.340
It's singular, right?
00:36:47.340 --> 00:36:50.480
It's got a singular matrix
there in the middle,
00:36:50.480 --> 00:36:56.110
and, let's see, so,
OK, it's singular,
00:36:56.110 --> 00:37:01.870
maybe you can tell me, its rank?
00:37:01.870 --> 00:37:04.320
What's the rank of A?
00:37:04.320 --> 00:37:08.900
It's clearly --
somebody just say it --
00:37:08.900 --> 00:37:09.920
one, thanks.
00:37:09.920 --> 00:37:15.160
The rank is one,
so the null space,
00:37:15.160 --> 00:37:18.850
what's the dimension
of the null space?
00:37:18.850 --> 00:37:19.890
One.
00:37:19.890 --> 00:37:20.390
Right?
00:37:20.390 --> 00:37:23.940
We've got a two-by-two
matrix of rank one,
00:37:23.940 --> 00:37:26.820
so of all that stuff from
the beginning of the course
00:37:26.820 --> 00:37:30.310
is still with us.
00:37:30.310 --> 00:37:32.790
The dimensions of those
fundamental spaces
00:37:32.790 --> 00:37:36.950
is still central,
and a basis for them.
00:37:36.950 --> 00:37:40.890
Now, can you tell me a vector
that's in the null space?
00:37:40.890 --> 00:37:47.000
And then that will be my last
point to make about the SVD.
00:37:47.000 --> 00:37:49.400
Can you tell me a vector
that's in the null space?
00:37:54.410 --> 00:38:00.820
So what would I multiply
by and get zero, here?
00:38:00.820 --> 00:38:04.120
I think the answer
is probably v2.
00:38:04.120 --> 00:38:07.820
I think probably v2
is in the null space,
00:38:07.820 --> 00:38:11.950
because I think that must
be the eigenvector going
00:38:11.950 --> 00:38:14.800
with this zero eigenvalue.
00:38:14.800 --> 00:38:16.310
Yes.
00:38:16.310 --> 00:38:17.200
Have a look at that.
00:38:17.200 --> 00:38:21.390
And I could ask you the
null space of A transpose.
00:38:21.390 --> 00:38:23.050
And I could ask you
the column space.
00:38:23.050 --> 00:38:24.620
All that stuff.
00:38:24.620 --> 00:38:27.180
Everything is sitting
there in the SVD.
00:38:27.180 --> 00:38:29.990
The SVD takes a little
more time to compute,
00:38:29.990 --> 00:38:36.090
but it displays all the
good stuff about a matrix.
00:38:36.090 --> 00:38:36.720
OK.
00:38:36.720 --> 00:38:39.680
Any question about the SVD?
00:38:39.680 --> 00:38:47.800
Let me keep going
with further topics.
00:38:47.800 --> 00:38:48.990
Now, let's see.
00:38:48.990 --> 00:38:51.050
Similar matrices
we've talked about,
00:38:51.050 --> 00:38:55.390
let me see if I've
got another, --
00:38:55.390 --> 00:38:57.620
OK.
00:38:57.620 --> 00:39:04.570
Here's a true false, so
we can do that, easily.
00:39:04.570 --> 00:39:05.300
So.
00:39:05.300 --> 00:39:07.560
Question, A given.
00:39:10.480 --> 00:39:17.840
A is symmetric and orthogonal.
00:39:20.861 --> 00:39:21.360
OK.
00:39:26.920 --> 00:39:29.870
So beautiful matrices like that
don't come along every day.
00:39:29.870 --> 00:39:36.480
But what can we say first
about its eigenvalues?
00:39:36.480 --> 00:39:38.680
Actually, of course.
00:39:38.680 --> 00:39:41.810
Here are our two most
important classes of matrices,
00:39:41.810 --> 00:39:45.500
and we're looking
at the intersection.
00:39:45.500 --> 00:39:48.470
So those really
are neat matrices,
00:39:48.470 --> 00:39:50.920
and what can you tell
me about what could
00:39:50.920 --> 00:39:52.670
the possible eigenvalues be?
00:39:52.670 --> 00:39:57.540
Eigenvalues can be what?
00:39:57.540 --> 00:39:59.330
What do I know about
the eigenvalues
00:39:59.330 --> 00:40:01.280
of a symmetric matrix?
00:40:01.280 --> 00:40:04.690
Lambda is real.
00:40:04.690 --> 00:40:06.550
What do I know about
the eigenvalues
00:40:06.550 --> 00:40:10.260
of an orthogonal matrix?
00:40:10.260 --> 00:40:12.370
Ha.
00:40:12.370 --> 00:40:13.200
Maybe nothing.
00:40:13.200 --> 00:40:15.420
But, no, that can't be.
00:40:15.420 --> 00:40:17.911
What do I know about the
eigenvalues of an orthogonal
00:40:17.911 --> 00:40:18.410
matrix?
00:40:18.410 --> 00:40:19.565
Well, what feels right?
00:40:22.330 --> 00:40:26.620
Basing mathematics on just
a little gut instinct here,
00:40:26.620 --> 00:40:29.910
the eigenvalues of
an orthogonal matrix
00:40:29.910 --> 00:40:33.800
ought to have magnitude one.
00:40:33.800 --> 00:40:36.280
Orthogonal matrices
are like rotations,
00:40:36.280 --> 00:40:40.921
they're not changing the length,
so orthogonal, the eigenvalues
00:40:40.921 --> 00:40:41.420
are one.
00:40:41.420 --> 00:40:50.070
Let me just show you why.
00:40:50.070 --> 00:40:53.920
So the matrix, can I
call it Q for orthogonal
00:40:53.920 --> 00:40:55.810
Why? for the moment?
00:40:55.810 --> 00:40:58.760
If I look at Q x
equal lambda x, how
00:40:58.760 --> 00:41:03.570
do I see that this
thing has magnitude one?
00:41:03.570 --> 00:41:06.140
I take the length of both sides.
00:41:06.140 --> 00:41:08.930
This is taking lengths,
taking lengths,
00:41:08.930 --> 00:41:13.520
this is whatever the magnitude
is times the length of x.
00:41:13.520 --> 00:41:18.170
And what's the length of Q x
if Q is an orthogonal matrix?
00:41:18.170 --> 00:41:20.910
This is something
you should know.
00:41:20.910 --> 00:41:23.100
It's the same as
the length of x.
00:41:23.100 --> 00:41:26.400
Orthogonal matrices
don't change lengths.
00:41:26.400 --> 00:41:30.330
So lambda has to be one.
00:41:30.330 --> 00:41:31.090
Right.
00:41:31.090 --> 00:41:31.890
OK.
00:41:31.890 --> 00:41:34.440
That's worth
committing to memory,
00:41:34.440 --> 00:41:36.930
that could show up again.
00:41:36.930 --> 00:41:37.430
OK.
00:41:37.430 --> 00:41:40.720
So what's the answer
now to this question,
00:41:40.720 --> 00:41:42.940
what can the eigenvalues be?
00:41:42.940 --> 00:41:45.220
There's only two
possibilities, and they
00:41:45.220 --> 00:41:55.050
are one and the other
one, the other possibility
00:41:55.050 --> 00:42:00.770
is negative one, right, because
these have the right magnitude,
00:42:00.770 --> 00:42:03.090
and they're real.
00:42:03.090 --> 00:42:04.070
OK.
00:42:04.070 --> 00:42:04.730
TK.
00:42:04.730 --> 00:42:07.090
true -- OK.
00:42:07.090 --> 00:42:08.260
True or false?
00:42:08.260 --> 00:42:12.220
A is sure to be
positive definite.
00:42:12.220 --> 00:42:14.070
Well, this is a great
matrix, but is it
00:42:14.070 --> 00:42:16.570
sure to be positive definite?
00:42:16.570 --> 00:42:17.090
No.
00:42:17.090 --> 00:42:19.380
If it could have an
eigenvalue minus one,
00:42:19.380 --> 00:42:21.910
it wouldn't be
positive definite.
00:42:21.910 --> 00:42:24.230
True or false, it has
no repeated eigenvalues.
00:42:27.820 --> 00:42:29.700
That's false, too.
00:42:29.700 --> 00:42:31.710
In fact, it's going to
have repeated eigenvalues
00:42:31.710 --> 00:42:33.740
if it's as big as
three by three,
00:42:33.740 --> 00:42:35.630
one of these c- one
of these, at least,
00:42:35.630 --> 00:42:37.270
will have to get repeated.
00:42:37.270 --> 00:42:37.770
Sure.
00:42:37.770 --> 00:42:40.040
So it's got repeated
eigenvalues, but,
00:42:40.040 --> 00:42:42.910
is it diagonalizable?
00:42:42.910 --> 00:42:45.210
It's got these many, many,
repeated eigenvalues.
00:42:45.210 --> 00:42:46.900
If it's fifty by
fifty, it's certainly
00:42:46.900 --> 00:42:48.790
got a lot of repetitions.
00:42:48.790 --> 00:42:51.370
Is it diagonalizable?
00:42:51.370 --> 00:42:52.110
Yes.
00:42:52.110 --> 00:42:55.500
All symmetric matrices,
all orthogonal matrices
00:42:55.500 --> 00:42:57.090
can be diagonalized.
00:42:57.090 --> 00:43:02.620
And, in fact, the eigenvectors
can even be chosen orthogonal.
00:43:02.620 --> 00:43:05.120
So it could be, sort
of, like, diagonalized
00:43:05.120 --> 00:43:09.270
the best way with a Q,
and not just any old S.
00:43:09.270 --> 00:43:09.840
OK.
00:43:09.840 --> 00:43:11.820
Is it non-singular?
00:43:11.820 --> 00:43:15.275
Is a symmetric orthogonal
matrix non-singular?
00:43:19.500 --> 00:43:21.790
Orthogonal matrices are
always non-singular.
00:43:21.790 --> 00:43:22.900
Sure.
00:43:22.900 --> 00:43:26.910
And, obviously, we don't
have any zero Eigenvalues.
00:43:26.910 --> 00:43:28.670
Is it sure to be diagonalizable?
00:43:28.670 --> 00:43:31.310
Yes.
00:43:31.310 --> 00:43:41.370
Now, here's a final step -- show
that one-half of A plus I is A
00:43:41.370 --> 00:43:42.140
--
00:43:42.140 --> 00:43:51.705
that is, prove one-half of A
plus I is a projection matrix.
00:43:58.880 --> 00:43:59.380
OK?
00:44:04.080 --> 00:44:04.750
Let's see.
00:44:04.750 --> 00:44:05.510
What do I do?
00:44:09.170 --> 00:44:11.610
I could see two ways to do this.
00:44:11.610 --> 00:44:14.560
I could check the properties
of a projection matrix, which
00:44:14.560 --> 00:44:15.700
are what?
00:44:15.700 --> 00:44:18.520
A projection matrix
is symmetric.
00:44:18.520 --> 00:44:21.790
Well, that's certainly
symmetric, because A is.
00:44:21.790 --> 00:44:24.440
And what's the other property?
00:44:24.440 --> 00:44:26.230
I should square
it, and hopefully
00:44:26.230 --> 00:44:27.900
get the same thing back.
00:44:27.900 --> 00:44:32.230
So can I do that, square and see
if I get the same thing back?
00:44:32.230 --> 00:44:37.060
So if I square it, I'll get
one-quarter of A squared
00:44:37.060 --> 00:44:40.640
plus two A plus I, right?
00:44:40.640 --> 00:44:48.430
And the question is, does that
agree with p- the thing itself?
00:44:48.430 --> 00:44:53.480
One-half A plus I.
00:44:53.480 --> 00:44:53.980
Hm.
00:44:57.240 --> 00:45:02.060
I guess I'd like to know
something about A squared.
00:45:02.060 --> 00:45:03.260
What is A squared?
00:45:03.260 --> 00:45:05.090
That's our problem.
00:45:05.090 --> 00:45:06.260
What is A squared?
00:45:11.560 --> 00:45:13.510
If A is symmetric
and orthogonal,
00:45:13.510 --> 00:45:17.390
A is symmetric and orthogonal.
00:45:22.060 --> 00:45:23.710
This is what we're given, right?
00:45:23.710 --> 00:45:27.990
It's symmetric, and
it's orthogonal.
00:45:27.990 --> 00:45:31.350
So what's A squared?
00:45:31.350 --> 00:45:36.360
I. A squared is I,
because A times A --
00:45:36.360 --> 00:45:42.500
if A equals its own inverse,
so A times A is the same as A
00:45:42.500 --> 00:45:46.150
times A inverse, which is I.
00:45:46.150 --> 00:45:52.320
So this A squared here is I.
00:45:52.320 --> 00:45:54.530
And now we've got it.
00:45:54.530 --> 00:45:57.810
We've got two identities
over four, that's good,
00:45:57.810 --> 00:46:01.060
and we've got two As
over four, that's good.
00:46:01.060 --> 00:46:01.670
OK.
00:46:01.670 --> 00:46:05.960
So it turned out to be a
projection matrix safely.
00:46:05.960 --> 00:46:08.310
And we could also
have said, well,
00:46:08.310 --> 00:46:11.230
what are the eigenvalues
of this thing?
00:46:11.230 --> 00:46:14.870
What are the eigenvalues
of a half A plus I?
00:46:14.870 --> 00:46:17.970
If the eigenvalues of A
are one and minus one,
00:46:17.970 --> 00:46:21.840
what are the
eigenvalues of A plus I?
00:46:21.840 --> 00:46:25.860
Just stay with it these
last thirty seconds here.
00:46:25.860 --> 00:46:29.140
What if I know these
eigenvalues of A,
00:46:29.140 --> 00:46:31.480
and I add the identity,
the eigenvalues
00:46:31.480 --> 00:46:35.600
of A plus I are zero and two.
00:46:35.600 --> 00:46:39.480
And then when I divide by two,
the eigenvalues are zero and
00:46:39.480 --> 00:46:40.020
one.
00:46:40.020 --> 00:46:43.130
So it's symmetric, it's
got the right eigenvalues,
00:46:43.130 --> 00:46:45.070
it's a projection matrix.
00:46:45.070 --> 00:46:49.290
OK, you're seeing a lot of
stuff about eigenvalues,
00:46:49.290 --> 00:46:54.460
and special matrices, and
that's what the quiz is about.
00:46:54.460 --> 00:46:57.230
OK, so good luck on the quiz.